Decomposition Operator (↓)

Formal Definition

The decomposition operator (↓) breaks down complex skills into their constituent subskills. It is the complementary operation to composition, enabling analysis and understanding of skill structure.

Formally, the decomposition operator is defined as:

${\mathcal{S}}_{complex}\downarrow ={\mathcal{M}}_d({\mathcal{S}}_{complex})=\{{\mathcal{S}}_1,{\mathcal{S}}_2,\dots ,{\mathcal{S}}_n\}$

Where:

• ${\mathcal{S}}_{complex}\in \mathcal{S}$: The complex skill to decompose
• ${\mathcal{M}}_d$: Decomposition metaskill
• Result: A set of subskills $\{{\mathcal{S}}_1,{\mathcal{S}}_2,\dots ,{\mathcal{S}}_n\}$

Decomposition Constraint

The decomposed subskills should approximately reconstruct the original skill:

${◯}_{i=1}^n{\mathcal{S}}_i\approx {\mathcal{S}}_{complex}$

Where $◯$ represents composition under an appropriate composition type.

Key Properties and Characteristics

1. Decomposition Completeness (Axiom 2)

Every complex skill has a valid decomposition: $\forall {\mathcal{S}}_{complex}\in \mathcal{S},\exists \{{\mathcal{S}}_1,\dots ,{\mathcal{S}}_n\}\subseteq \mathcal{S}:{\mathcal{S}}_{complex}\downarrow =\{{\mathcal{S}}_1,\dots ,{\mathcal{S}}_n\}$

2. Non-Uniqueness

Decomposition may not be unique - a skill can be broken down in multiple valid ways: ${\mathcal{S}}_{complex}\downarrow =\{{\mathcal{S}}_1,{\mathcal{S}}_2\}\text{ or }\{{\mathcal{S}}_3,{\mathcal{S}}_4,{\mathcal{S}}_5\}$

3. Granularity

The level of decomposition can vary based on context and requirements.

4. Reconstruction Fidelity

Ideally, recomposing subskills should yield the original skill (or a close approximation).

Research Context and Applications

Decomposition is essential for understanding and improving skills:

• Skill Analysis: Understanding the components of complex capabilities
• Learning Path Design: Identifying what subskills need to be learned
• Debugging: Isolating which component skill is underperforming
• Transfer Learning: Identifying reusable subskills across tasks
• Curriculum Learning: Sequencing learning from simple to complex

In LLM research:

• Breaking down complex prompts into sub-tasks
• Analyzing which capabilities contribute to performance
• Identifying skill gaps for targeted training
• Designing modular prompt structures

Connections to Other Concepts

• Skills (𝒮): Elements being decomposed
• Subskills (𝒮_sub): Results of decomposition
• Decomposition Metaskill (${\mathcal{M}}_d$): Implements the decomposition operator
• Composition Operator (∘): Complementary operation that combines skills
• Superskills: Skills that can be decomposed into subskills
• Skill Hierarchy: Decomposition reveals hierarchical structure

Decomposition Strategies

Different strategies can be employed for decomposition:

1. Functional Decomposition: Break down by functional components
2. Sequential Decomposition: Identify temporal sequence of subskills
3. Domain Decomposition: Separate by domain-specific capabilities
4. Abstraction Levels: Decompose by levels of abstraction

Open Research Questions

1. Uniqueness: Under what conditions is decomposition unique?
2. Optimal Granularity: What is the optimal level of decomposition for a given context?
3. Automated Decomposition: How can decomposition be automated for LLMs?
4. Minimal Decomposition: What is the minimal complete decomposition of a skill?
5. Decomposition Metrics: How to measure the quality of a decomposition?
6. Recursive Decomposition: When should decomposition stop (base skills)?
7. Context Sensitivity: How does context affect decomposition strategy?